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    <title>systems</title>
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    <center>Scilab Function</center>
    <div align="right">Last update : April 1993</div>
    <p>
      <b>systems</b> -  a collection of dynamical system</p>
    <h3>
      <font color="blue">Calling Sequence</font>
    </h3>
    <dl>
      <dd>
        <tt>[]=systems()  </tt>
      </dd>
    </dl>
    <h3>
      <font color="blue">Description</font>
    </h3>
    <p>
    A call to this function will load into Scilab a set of macros which
    describes dynamical systems. Their parameters can be initiated by 
    calling the routine tdinit().
  </p>
    <h3>
      <font color="blue">Bioreact</font>
    </h3>
    <dl>
      <pre>

[ydot]=biorecat(t,x)
   
    </pre>
      <p>
    a bioreactor model, 
  </p>
      <dd>
        <li>
          <b>
            <font color="maroon">x(1)</font>
          </b>is the biomass concentration</li>
        <li>
          <b>
            <font color="maroon">x(2)</font>
          </b>is the sugar concentration</li>
      </dd>
      <pre>

                    xdot(1)=mu_td(x(2))*x(1)- debit*x(1);
                    xdot(2)=-k*mu_td(x(2))*x(1)-debit*x(2)+debit*x2in;
   
    </pre>
      <p>
    where mu_td is given by 
  </p>
      <pre>

                    mu_td(x)=x/(1+x);
   
    </pre>
    </dl>
    <h3>
      <font color="blue">Compet</font>
    </h3>
    <dl>
      <pre>

[xdot]=compet(t,x [,u])
   
    </pre>
      <p>
    a competition model. <tt>
          <b>x(1),x(2)</b>
        </tt> stands for two populations which  grows on a same resource. <tt>
          <b>1/u</b>
        </tt> is the level of that resource ( 1 is the default value).
  </p>
      <pre>

xdot=0*ones(2,1);
xdot(1) = ppr*x(1)*(1-x(1)/ppk) - u*ppa*x(1)*x(2) ,
xdot(2) = pps*x(2)*(1-x(2)/ppl) - u*ppb*x(1)*x(2) ,
   
    </pre>
      <p>
     "The macro <tt>
          <b>[xe]=equilcom(ue)</b>
        </tt>" computes an equilibrium point of the competition model and a fixed  level of the resource ue ( default value is 1)
  </p>
      <p>
     "The macro <tt>
          <b>[f,g,h,linsy]=lincomp([ue])</b>
        </tt>" gives the linearisation of the competition model ( with output y=x) around the equilibrium point xe=equilcom(ue). This macro returns [f,g,h] the three matrices of the linearised system. and linsy which is a Scilab macro [ydot]=linsy(t,x) which computes the  dynamics of the linearised system
  </p>
    </dl>
    <h3>
      <font color="blue">Cycllim</font>
    </h3>
    <dl>
      <pre>

[xdot]=cycllim(t,x)
   
    </pre>
      <p>
    a model with a limit cycle 
  </p>
      <pre>

  xdot=a*x+qeps(1-||x||**2)x
   
    </pre>
    </dl>
    <h3>
      <font color="blue">Linear</font>
    </h3>
    <dl>
      <pre>

[xdot]=linear(t,x)
   
    </pre>
      <p>
    a linear system 
  </p>
    </dl>
    <h3>
      <font color="blue">Blinper</font>
    </h3>
    <dl>
      <pre>

[xdot]=linper(t,x)
   
    </pre>
      <p>
    a linear system with quadratic perturbations.
  </p>
    </dl>
    <h3>
      <font color="blue">Pop</font>
    </h3>
    <dl>
      <pre>

[xdot]=pop(t,x)
   
    </pre>
      <p>
    a fish population model
  </p>
      <pre>

xdot= 10*x*(1-x/K)- peche(t)*x
   
    </pre>
    </dl>
    <h3>
      <font color="blue">Proie</font>
    </h3>
    <dl>
      <p>
    a Predator prey model with external insecticide.
  </p>
      <pre>

[xdot]=p_p(t,x,[u]
   
    </pre>
      <dd>
        <li>
          <b>
            <font color="maroon">x(1)</font>
          </b>The prey ( that we want to kill )</li>
        <li>
          <b>
            <font color="maroon">x(2)</font>
          </b>the predator ( that we want to preserve )</li>
        <li>
          <b>
            <font color="maroon">u</font>
          </b>mortality rate due to insecticide which  destroys both prey and predator ( default value u=0)</li>
      </dd>
      <pre>

xdot(1) = ppr*x(1)*(1-x(1)/ppk) - ppa*x(1)*x(2) - u*x(1);
xdot(2) = -ppm*x(2)             + ppb*x(1)*x(2) - u*x(2);
   
    </pre>
      <p>
     The macro <tt>
          <b>[xe]=equilpp([ue])</b>
        </tt> computes the equilibrium point of the <tt>
          <b>p_p</b>
        </tt> system for the value <tt>
          <b>ue</b>
        </tt> of the command. The default value for <tt>
          <b>ue</b>
        </tt> is 0.</p>
      <pre>

                    xe(1) =  (ppm+ue)/ppb;
                    xe(2) =  (ppr*(1-xe(1)/ppk)-ue)/ppa;
   
    </pre>
    </dl>
    <h3>
      <font color="blue">Lincom</font>
    </h3>
    <dl>
      <pre>

[xdot]=lincom(t,x,k)
   
    </pre>
      <p>
    linear system with a feedback</p>
      <pre>

 xdot= a*x +b*(-k*x)
   
    </pre>
    </dl>
    <h3>
      <font color="blue">See Also</font>
    </h3>
    <p>
      <a href="tdinit.htm">
        <tt>
          <b>tdinit</b>
        </tt>
      </a>,&nbsp;&nbsp;</p>
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